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Dr. Katz DEq Homework Solutions 95

Dr. Katz DEq Homework Solutions 95 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 95 To conclude, we simply observe that as c-ar c-br the inequalities in (6.6.0.15) and (6.6.0.16), already established, are neces- sarily strict. This concludes the proof of (6.6.0). (6.6.1) Combining 6.3 and 6.6.0, we are "reduced" to proving the implication (6.2.6)~ (6.2.4) under the additional hypothesis that the rational numbers a, b, c satisfy (6.6.1.0) aCZ, b6Z, c~Z, c-aCZ, c-bc~Z, a-b6Z, c-a-b~Z. (6.6.2) Corollary. In order that rational numbers a, b, c satisfy (6.2.6) and (6.6.1.0), it is necessary and sufficient that for any A eZ which is invertible modulo N for a common denominator N of a, b, c, the rational numbers A a, A b, A c satisfy (6.2.6) and (6.6.1.0). Proof The sufficiency is clear; take A = 1. For the necessity, (6.6.1.0) will still hold because d is invertible mod N, and the condition (6.6.0.2) is obviously invariant under (a, b, c)--* (A a, A b, A c) for such A. By (6.6.0), (6.6.0.2) and (6.6.1.0) imply (6.2.6). (6.6.3) Corollary. In order that rational numbers a, b, c satisfy (6.2.6) and (6.6.1.0), it is necessary and sufficient that for any integers r, s, t, the
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